A Globally Convergent Successive Approximation Method for Severely Nonsmooth Equations

This paper presents a globally convergent successive approximation method for solving $F(x)=0$ where $F$ is a continuous function. At each step of the method, $F$ is approximated by a smooth function $f_{k},$ with $\pa f_{k}-F\pa \rightarrow 0$ as $k \rightarrow \infty$. The direction $-f'_{k}(x_{k})^{-1}F(x_{k})$ is then used in a line search on a sum of squares objective. The approximate function $f_k$ can be constructed for nonsmooth equations arising from variational inequalities, maximal monotone operator problems, nonlinear complementarity problems, and nonsmooth partial differential equations. Numerical examples are given to illustrate the method.